Find the Inverse [[1,-2,0],[0,0,1],[0,1,0]]

Problem

[[1,−2,0],[0,0,1],[0,1,0]]

Solution

1. Set up the augmented matrix by placing the identity matrix I to the right of the given matrix A

[[1,−2,0,|,1,0,0],[0,0,1,|,0,1,0],[0,1,0,|,0,0,1]]

2. Swap the second and third rows ((R_2)↔(R_3) to move the pivot into the correct position for row-echelon form.

[[1,−2,0,|,1,0,0],[0,1,0,|,0,0,1],[0,0,1,|,0,1,0]]

3. Eliminate the −2 in the first row by adding twice the second row to the first row ((R_1)→(R_1)+2*(R_2).

[[1,0,0,|,1,0,2],[0,1,0,|,0,0,1],[0,0,1,|,0,1,0]]

4. Identify the resulting matrix on the right side as the inverse matrix A(−1)

[[1,0,2],[0,0,1],[0,1,0]]

Final Answer

[[1,−2,0],[0,0,1],[0,1,0]](−1)=[[1,0,2],[0,0,1],[0,1,0]]

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